integral of (2x+3)/(sqrt(9-8x-x^2))

Lösung

\int \frac{2 x + 3}{9 - 8 x - x ^{2}} d x

- 2 - x^{2} - 8 x + 9 - 5 arcsin (\frac{1}{5} (x + 4)) + C

Schritte zur Lösung

\int \frac{2 x + 3}{9 - 8 x - x ^{2}} d x

Vervollständige das Quadrat

9 - 8 x - x^{2} : - (x + 4)^{2} + 25

= \int \frac{2 x + 3}{- ( x + 4 ) ^{2} + 25} d x

Wende U-Substitution an

= \int \frac{2 u - 5}{- u ^{2} + 25} d u

Multipliziere aus

\frac{2 u - 5}{- u ^{2} + 25} : \frac{2 u}{- u ^{2} + 25} - \frac{5}{- u ^{2} + 25}

= \int \frac{2 u}{- u ^{2} + 25} - \frac{5}{- u ^{2} + 25} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{2 u}{- u ^{2} + 25} d u - \int \frac{5}{- u ^{2} + 25} d u

\int \frac{2 u}{- u ^{2} + 25} d u = - 2 - u^{2} + 25

\int \frac{5}{- u ^{2} + 25} d u = 5 arcsin (\frac{1}{5} u)

= - 2 - u^{2} + 25 - 5 arcsin (\frac{1}{5} u)

Setze in

u = x + 4

ein

= - 2 - (x + 4)^{2} + 25 - 5 arcsin (\frac{1}{5} (x + 4))

Multipliziere aus

- (x + 4)^{2} + 25 : - x^{2} - 8 x + 9

= - 2 - x^{2} - 8 x + 9 - 5 arcsin (\frac{1}{5} (x + 4))

Füge eine Konstante zur Lösung hinzu

= - 2 - x^{2} - 8 x + 9 - 5 arcsin (\frac{1}{5} (x + 4)) + C

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